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Re: Grist for layering discussion

From: Pat Hayes <phayes@ai.uwf.edu>
Date: Tue, 15 Jan 2002 16:06:05 -0600
Message-Id: <p0510103db86a5356cc1e@[65.212.118.208]>
To: "Tim Berners-Lee" <timbl@w3.org>
Cc: <pfps@research.bell-labs.com>, <sandro@w3.org>, <phayes@ai.uwf.edu>, <hendler@cs.umd.edu>, <las@olin.edu>, <connolly@w3.org>, <w3c-semweb-ad@w3.org>, <www-archive@w3.org>
>----- Original Message -----
>From: "Peter F. Patel-Schneider" <pfps@research.bell-labs.com>
>To: <timbl@w3.org>
>Cc: <sandro@w3.org>; <phayes@ai.uwf.edu>; <hendler@cs.umd.edu>;
><las@olin.edu>; <connolly@w3.org>; <w3c-semweb-ad@w3.org>;
><www-archive@w3.org>
>Sent: Saturday, January 12, 2002 10:17 AM
>Subject: Re: Grist for layering discussion
>
>
>>  I have been guilty of misrepresenting the law of the excluded middle.
>>
>>  The Law of the Excluded Middle states that any proposition is either true
>>  or false, i.e., no other possibilities exist.  There are several logical
>>  systems that do not have the Law of the Excluded Middle.  These logics, in
>>  essense, add one, or more, other possible situations for a proposition.
>>  Some such logics add an ``undefined''; some add a ``contradictory''; some
>>  add both; some add even more.
>
>I understand from Lynn that basically any finite number of states
>leaves you with the same problem. In other words, if the "undefined"
>is a state you can deduce something must be in by a process of
>elimination then it allows you to construct a paradox.  These seem to be
>systems in which the middle is still excluded there are just more things
>outside it.  To not not have the PEM at all, you have to limit the
>inference so that there are things which, while in fact are not true
>or false, you don't have a label for what they are, or axioms for
>that label which allow you to hang yourself.
>
>"This statement has truth-value undefined".

Right, exactly. The 'weak' 3-valued logics do this, but they are then 
so weak that you can hardly do anything with them. They act like 
2-valued logics but if anything is undefined then all assertions just 
vanish into undefinedness. (They actually model computation quite 
faithfully in a sense, if you interpret 'undefined' as 'no value gets 
returned', ie 'wanders into weeds', in Sandro's phrase.Then if 
anything subexpression wanders into the weeds, you never get a value. 
But what you can't ever do is *conclude* that something is in the 
weeds.)

>  >Some logics even abandon the whole idea of
>>  propositions have a truth value.
>
>Well, I am sure ... though without knowing what replaces it
>i don't know how to react.   But for example on the sweb there  is
>no absolute truth, there is just the starting place a process takes
>for its reasoning.

Yes, but that is what 'truth' means in logic as well. Don't think 
that not knowing whether something is 'really' true implies that one 
has to abandon 2-valued logic. The logic only says, in effect: IF 
these are true, THEN this other stuff Ive deduced from it must be 
true as well.

>
>>  From: Peter F. Patel-Schneider <pfps@research.bell-labs.com>
>>  Subject: Re: Grist for layering discussion
>>  Date: Fri, 11 Jan 2002 20:14:08 -0500
>>
>>  > > The normal way you do it relies on the Princple of the Excluded
>middle,
>>  > > which of course cannot be part of the semantic web.
>>  >
>>  > Why not?  Well, of course, I don't want to deduce anything just because
>>  > some web page says X and some other web page says not X, but there are
>lots of
>>  > formalisms that don't have this property and that have the Law of the
>>  > Excluded Middle.
>
>Why not? Because you don't want to deduce anything be *contemplating*
>a paradox.  If you start off with a paradox in your inital data (stuff taken
>as
>true for this run) then you will of course run amok, as you would had you
>started
>simply with a falsehood. But I had rules aout the law of the excluded middle
>in that I want a system which can't be crashed by asking it to consider
>a hypothetical paradox.  ("Consider "this is false" - it must either be true
>or
>false, right?")

Good example. Its a bad idea to even allow things like that to be 
said, because the task of telling whether some larger DB implies one 
of them isn't computable, in general, so you can never be sure you 
have filtered them out. If paradoxes - not contradictions like (p and 
not p), which are always false, but *paradoxes* - which can't 
rationally be either true or false - can be stated in the language, 
then all reasoning from any source is suspect. And (again unlike a 
contradiction) the problem might not show up in any way you can 
detect. You just have incompatible conclusions where each of them 
might have impeccable and checkable derivations, both warranted 
correct, but disagreeing. Not a good thing to have.

>
>>  I should have gone on and said:
>>
>>  There are logics (including most of the first group above) that don't have
>>  the Law of the Excluded Middle but nonetheless would be able to deduce
>>  anything from an explicit contradiction.
>
>I don't see a problem with being able to deduce anything from
>an explicit contradiction in what one belives as true.

Sure, but a paradox is worse than a contradiction. Contradictions are 
orcs, but a paradox is a balrog.

>  >  There are other logics (including
>>  most of the second group above) where even explicit contradictions don't
>>  enable the inference of all propositions.
>>
>>  You can't claim that abandoning the Law of the Excluded Middle solves a
>>  particular problem without saying what you are going to replace it by.
>
>What we seem to be using, as Dan mentioned, is a form of not in which
>certain
>properties have converses, and some (most normal ones) don't.   Basically,
>you have to show how to something does before anyone can assume so.

That sounds very like intuitionist logic, or maybe some other 
constructive logic. The key idea for the intuitionists was that to 
prove not-P, it wasn't enough to just demonstrate that assuming P 
produces a contradiction. You had to actually prove the negation 
*directly*. It amounts to a restriction on kinds of proofs that are 
allowed.

Pat
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Received on Tuesday, 15 January 2002 17:05:04 GMT

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